A Combinatorial Approach to Planar Non-colliding Robot Arm Motion Planning Ileana Streinu
Department of Computer Science Smith College Northampton, MA 01063, USA
[email protected] http://cs.smith.edu/
Abstract
We propose a combinatorial approach to plan noncolliding motions for a polygonal bar-and-joint framework. Our approach yields very eÆcient deterministic algorithms for a category of robot arm motion planning problems with many degrees of freedom, where the known general roadmap techniques would give exponential complexity. It is based on a novel class of one-degree-of-freedom mechanisms induced by pseudo triangulations of planar point sets, for which we provide several equivalent characterization and exhibit rich combinatorial and rigidity theoretic properties. The main application is an eÆcient algorithm for the Carpenter's Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non self-intersecting planar motions. A step in the convexi cation motion consists in moving a pseudotriangulation-based mechanism along its unique trajectory in con guration space until two adjacent edges align. At that point, a local alteration restores the pseudo triangulation. The motion continues for O(n2 ) steps until all the points are in convex position. 1 Introduction
In this paper we present a new approach to a category of planar motion planning problems, including noncolliding unfoldings of open and closed chains (robot arms), even in the presence of certain types of obstacles (movable convex polygons or exible polygonal chains). All of our constructions are elementary and are based on a novel class of planar embedded graphs called pseudo-triangulations which possess rich combinatorial properties. We use techniques from Rigid Supported
by NSF grant CCR-9731804.
streinu
ity Theory and Visibility graphs, together with insights gained from Oriented Matroid Theory. The main ideas of our approach are best illustrated by the basic problem of continuously recon guring a simple planar polygon to any other planar con guration with the same edge-lengths, while remaining in the plane and without creating self-intersections along the way, for which we will describe a simple and eÆcient algorithm. This is done by rst nding motions that convexify both con gurations with monotone motions (de ned below), then taking one path in reverse. The paper is organized as follows. In the remaining of this introduction, we give an informal high-level view of the result and its connection with previous work. Preliminary concepts are de ned in section 2. Section 3 contains several combinatorial characterizations of pseudo triangulations, as well as a sketch of algorithms for computing them. Section 4 contains the rigidity theoretic results on pseudo-triangulations. Section 5 contains the description of the global convexi cation motion and complexity analysis. We conclude with some suggestions for further research. Frameworks and Robot arms. A bar-and-joint framework is a combinatorial graph G = (V; E ) embedded in the plane with rigid bars ( xed length straight line segments) corresponding to the edges. Edges can move freely in the plane around adjacent joints (vertices). The motions preserve rigidly the lengths of the graph edges, but impose no restriction on the non-edges, which may increase or decrease freely. In general, edges may cross and slide over each other during the motion, but in this paper we are interested in avoiding collisions and will not allow this. Of particular interest are the monotone motions, where all the pairwise interdistances between vertices never decrease during the motion, thus guaranteeing non-collision. A linkage or robot arm is a planar framework whose underlying graph is a non-self-intersecting path with n vertices, and a closed chain is a simple planar polygon on n vertices. Straightening a linkage means moving it continuously until all its vertices lie on one line with non-overlapping edges. Convexifying a chain means moving it to a position where it forms a simple con-
vex polygon. Other types of frameworks of interest in this paper include semi-simple polygons and pseudotriangulations, de ned below. The Carpenter's Rule Problem: Is it always possible to straighten a planar linkage, or to convexify a planar chain? This question has been open since the 1970's. Recently, Connelly, Demaine and Rote [12] have answered it in the aÆrmative. Their solution still left open the problem: Find, algorithmically, a nite sequence of simple ( nitely described) motions to straighten a linkage, or to convexify a polygon.
Previous Results on Recon guring Linkages.
The general techniques for solving motion planning problems based on roadmaps work well on problems with bounded degrees of freedom, but yield exponential algorithms for high degree of freedom problems such as the one we are interested in. See [32], [10], [4] and [5]. In practical applications, probabilistic roadmap heuristics were used instead, see [24], [18]. Under various conditions, problems about recon guration of linkages range in complexity from polynomial ([26]) to NP- and even PSPACE-hard, see [19], [38], [21]. The particular problem of straightening bar-andjoint linkages and convexifying polygons has accumulated a distinguished history, with some approaches going back to a question of Erdos [14]. See Toussaint [34] for a fascinating account. There are abundant connections with work done in the computational biology, chemistry and physics literature and motivated by topics such as protein folding or molecular modeling. When crossings are allowed, Lenhart and Whitesides [26] have shown that the con guration space has at most two connected components and gave a linear algorithm for convexi cation based on simple motions moving only a constant number of joints at a time. Recent results in the mathematics literature [22] aim at understanding the topology of the con guration space of closed chains, but they allow crossings. The universality result for mechanical linkages ([20], [23]) holds for the general self-intersecting case. Studying recon gurations of linkages with non crossing motions has received a recent impetus in [28], and results on planar linkages using spatial motions ([7], [1]), trees, 3 and higher-dimensional linkages ([6], [11]) have followed. The Carpenter's Rule question, raised in the 1970's in the Topology community by G. Bergman, U. Grenander, S. Schanuel (cf. [25]) and independently in the early 1990's in the Computer Science community by W. Lenhart, S. Whitesides, J. Mitchell, seems to have rst appeared in print in [26] and [25]. It was recently settled by Connelly, Demaine and Rote [12]: all chains can be convexi ed, all linkages can be straightened. Their approach is to rst prove (using linear programming duality and Maxwell's theorem, using a technique originating in Crapo and Whiteley [13] and Whiteley [35]) that there always exists a local, in nitesimal motion which never decreases any interdistances. The actual velocities can be found using linear programming. Then they provide a global argument, showing the existence of a
continuous deformation obtained by integrating the resulting vector eld. Our Results. We strengthen and provide an algorithmic extension of the above mentioned result. While they have proven that the con guration space of planar non-crossing chains with the same orientation is connected, we show how to algorithmically compute a path, consisting of a nite number of nitelydescribed arcs, between any two such con gurations. Along the way, we characterize a family of planar rigid frameworks called pseudo-triangulations, which yield 1DOF monotone mechanisms when a convex hull edge is removed, a result of independent interest in Rigidity Theory. The rich combinatorial and rigidity-theoretic properties of pseudo-triangulations which we present in this paper are likely to add to the applicability of this versatile data structure, a slight generalization of the one introduced by Pocchiola and Vegter[29] in their study of the visibility complex and recently applied to kinetic geometric algorithms ([2], [3]). Novelty. Our approach is based on the idea of abstracting oriented-matroidal properties that hold throughout a portion of a continuous motion. The rigidity theoretic properties of our generalized pseudo triangulations are novel, as is the pseudo-triangulation based approach to 1DOF monotone mechanisms for nding a path in con guration space. Along the way, we also give a generalization of a key lemma in [12] by showing the non-existence of self-stress in bar-andstrut frameworks of a more complex structure than those arising from polygons.
Proof Techniques and Overview of the Convexi cation Algorithm. The convexifying path, seen in
the (2n 3)-dimensional con guration space (translations and rotations of the original chain being factored out), is a nite sequence of curve segments (arcs) connecting continuously at their endpoints. Each arc corresponds to the unique free motion of a monotone, one-degree-of-freeedom (1DOF) mechanism induced by a planar pseudo-triangulation with a convex hull edge removed. The mechanism is constructed algorithmically by adding n 4 bars to the original polygon. It is set in motion by pinning down an edge and rotating another edge around one of its joints. We show that this induces the whole framework to move monotonically, i.e. with non-decreasing interdistances between all pairs of vertices. One step of the convexi cation algorithm consists in moving this mechanism until two adjacent edges align, at which moment it ceases to be a pseudo-triangulation. We either freeze a joint (if the aligned edges belong to the polygon) and locally patch a pseudo-triangulation for a polygon with one less vertex, or otherwise perform a local ip of the added diagonals. There are many ways to construct the initial pseudotriangulation (e.g. using an adaptation of the greedy
ip algorithm of Pocchiloa and Vegter [29]) or to patch it at the alignment points. For the sake of the analysis, we use a canonical way which helps us to keep track of a global integer valued weight function f (n). Each
alignment step decreases f (n) by at least one unit. Hence the algorithm stops in at most f (n) such steps. The analysis of one of the convexi cation schemes would yield O(n2 ).
Combinatorial Rigidity and Pseudo triangulations. A 1DOF monotone mechanism obtained from
a pseudo-triangulation is an abstraction and a canonical representation of one of the many basic solutions, inducing monotone in nitesimal motions, that the linear programming approach of [12] would nd for a certain position of the polygon in its con guration space. We characterize pseudo-triangulations in several equivalent ways, exhibiting their rich combinatorial properties. Some of these are specialized versions of Laman's 2n 3 count and Henneberg constructions from combinatorial rigidity (see [36] or [17]). The proof of correctness of our approach derives from these properties, as well as from a generalization, from simple polygons to the wider class of pseudo-triangulation frameworks, of the approach used in [12] based on LP duality and Maxwell's theorem.
combinatorial properties of rigid frameworks. More precisely, a set of acyclic vectors has no linear combination with positive, not all zero coeÆcients that sums them to zero, while in the cyclic case there is always one. Collinearities may occur, see the examples in Fig. 1. The acyclic case of several collinear vectors but in the same direction, and the rest in acyclic relation with them will occur in our algorithm at the beginning of each step. A special cyclic case, which will occur at the end of each step of our convexi cation algorithm, will have some vectors collinear and in opposite directions, and all the others contained in a half-plane. An acyclic set of edges is a set of segments with endpoints in a nite set of planar points such that the vectors around each vertex, taken in the direction of the adjacent segments and directed away from the vertex, form an acyclic set. See Fig. 2.
2 De nitions and Preliminaries
1
References. For rigidity theory terminology and basic results, we refer the reader to [30], [36], [37] and [17]. In particular, rigidity, rst-order and generic rigidity, as well as classical results on 2-dimensional rigidity such as Laman's theorem, the Henneberg constructions, Lovasz and Yemini covering with two trees [27] and Maxwell's Theorem are to be found there. For oriented matroids, see [8], although we won't need more than the circular hyperline (or local) sequences of [9] and [16] (see also [33]). Notation and abbreviations. Our setting is the Euclidian plane. All index arithmetic is done mod n in the set [n] := f1; ; ng. We abbreviate \counterclockwise" as ccw and \one-degree-of-freedom mechanism" as 1DOF mechanism.
(b)
(a)
(d)
(e)
(c)
(a)
(b)
Figure 2: (a) An acyclic set of edges. Around each vertex, the vectors in the directions of the adjacent edges are depicted. (b) A set of edges which is cyclic, due to vertex 1.
Polygons, pseudo-triangles and pseudo k-gons. A polygon on the ordered set of points P = fp1 ; ; pn g
is obtained by joining pairs of consecutive points with line segments (edges of the polygon). It is simple if non-adjacent edges do not meet. In this case, there is a well-de ned and connected interior and exterior of the polygon. We will assume that the vertices are labeled in ccw order, i.e. such that the interior lies to the left when the boundary of the polygon is traversed in that order.
(f)
Figure 1: (a), (b) and (c) Acyclic and (d), (e) and (f) cyclic sets of vectors. (a)
Acyclic sets of vectors and edges. A set of vectors in R2 (with a common origin) is acyclic if it is strictly contained in a half-plane, and cyclic otherwise. The
terminology comes from oriented matroid theory (see [8]), an approach we won't explain in this abstract, but which provided the guiding line in our search for
(b)
(c)
Figure 3: (a) A pseudo-triangle. (b) A semi-simple pseudo-triangle. (c) A pseudo 4-gon and its two possible minimum pseudo-triangulations. A pseudo-triangle is a simple polygon with three vertices on its convex hull, joined by three inward
convex polygonal chains. In particular, a triangle is a pseudo-triangle. We introduce semi-simple pseudotriangles as a special case which allows for some degeneracies: some of the inner convex angles may be zero, but none of the inner re ex angles should be or 2 . More generally, if we focus on the convex vertices of a simple polygon and on the inner convex chains between them, we will refer to the polygon as being a pseudo k-gon if it has exactly k convex vertices. See Fig. 3(c) for an example. Semi-simple pseudo k-gons allow simple types of degeneracies: some of the k convex angles may be zero, but none of the re ex angles is or 2 . Note. A complete treatment of the general concept of semi-simple polygons as the limit case for simple polygons needs more technical apparatus and will not be covered in this paper. See Fig. 4 for examples and hints to some of the complexities involved in giving a complete de nition. 1
1
4
2
6
4 3
5
3
2
1 5
6
5 (b)
(a) 5
3
6
2
4
4
6
5
4
3 6
3
1
2
1
6 4
5 3
1
4
3 2
1
2 2
5
(c)
6
Figure 4: (a) A semi-simple and (b) a simple polygon on the same set of points, and (c) a semi-simple collinear polygon and several possible interpretations (and perturbations) of it. The main question is: which way could an opening motion go? For the main problem treated in this paper, and if we start with a simple polygon, then the only non simple polygons appearing along the way and which are needed to make our approach work will be semisimple k-gons.
(a)
(b)
(c)
Figure 5: (a) A minimum pseudo-triangulation. (b) A non-minimum pseudo-triangulation which contains a minimum one. (c) A non-minimum pseudo triangulation which does not contain a minimum pseudo triangulation.
Pseudo Triangulations. A pseudo-triangulation of a point set P is a partitioning of the convex hull of P into pseudo triangles such that every point appears
as the vertex of at least one pseudo triangular face. We can view it as a collection of edge segments with this property. Equivalently, it is a special embedding of a special planar graph such that the outer face is convex and all internal faces are pseudo triangles. A minimum pseudo triangulation is one which has the least number of edges among all pseudo triangulations of the same point set. See Fig. 5. Pseudo k-gons can also be further subdivided into smaller pseudo l-gons, l < k by the addition of inner diagonals. Call that a pseudo subdivision. E.g. a pseudo 4-gon can be partitioned into two (or more) pseudo triangles, see Fig.3(c). Frameworks. A framework (G; W ) is a graph G = (V; E ), jV j = n, together with a set of positive weights (lengths) W = fle je 2 E g. A realization of (G; W ) on a set pf points P = fp1 ; ; pn g is a mapping of vertices to points and edges to line segments (i.e. an embedding of G) so that the length of the segment corresponding to edge e is le . The set of all possible realizations of a framework is called its con guration space. As usual, we factor out translations and rotations. The con guration space may be empty, disconnected and in general has a complicated topological structure. The actual values of the edge lengths are not relevant to our discussion, hence by abuse of terminology, from now on we will refer to a realization (G; P ) as a framework. A motion or recon guration of the framework is a curve (one dimensional trajectory) in con guration space going through the point giving the framework realization.
(a)
(b)
Figure 6: The same graph embedded as two combinatorially distinct frameworks, which are 1DOF mechanisms: (a) is monotone, (b) is not. A combinatorial framework (G; M ) associated to a framework realization (G; P ) is obtained by retaining (in M ) only some combinatorial information from the underlying oriented matroid of the set of points P . Since in this paper we work only with special types of frameworks, we do not give here the general definition. In our particular case, the information M retained from the embedding will be, for each vertex, the signed circular sequence in which a directed line rotating ccw encounters the adjacent edge vectors. An edge vector is recorded positively or negatively depending on whether the rotating line encounters it in its direction or in the opposite one. This concept is a specialization of the local sequences of Goodman and Pollack [16] (see also [9] and [33]) and retains (partial) oriented matroid structure from the underlying set of
points P . In particular, we can read o from this information the planar nature of an embedding of the framework and the cyclic or acyclic nature of the edge vectors at each vertex. A framework is planar if its underlying graph is a plane graph (i.e. a planar graph plus the information about its face structure), and is embedded in a manner compatible with the plane graph structure (i.e. all faces are embedded as simple polygons). It is a pseudo-triangulation if all internal faces are pseudotriangles and the outer face is convex. A framework is acyclic if all the edge vectors around each vertex are acyclic. We also allow for the special situation (to the best of our knowledge, not dealt with in the literature on planar graph embeddings) when the faces of the embedded framework are semi-simple pseudo kgons (in particular semi-simple pseudo-triangles). For the careful reader who already noticed that some care must be exercized with the de nition of what it means for the outside face to be a semi-simple pseudo k-gon, we rush to add that it can be done in a natural way. First we have to show that the outer face is also some sort of pseudo k-gon. If the outer face is the convex hull of the points, when we \look from the outside" (i.e. from inside the outer face) it has no convex vertex, just a single re ex chain, hence it is an outer pseudo 0-gon. If it has a convex vertex (and hence one re ex chain) it is an outer pseudo 1-gon, etc. Notice that the outer pseudo 3-gon looks quite di erent from the usual pseudo triangle: its vertices are outside, not inside the convex hull of three convex vertices. For example, the outer face of the framework in Fig. 6(a) is an outer pseudo 1-gon, and for (b) is an outer 2-gon. In general, the outer face will be the only one with this special type of pseudo k-gon. Once we xed this detail, semi-simplicity is then extended in the obvious way.
(a)
is called a combinatorial pseudo triangulation. The underlying combinatorial framework captures the information (and can be used to represent) the combinatorial pseudo triangulation. A crucial idea in our convexi cation algorithm will be to use mechanisms obtained from pseudo triangulations, and to recon gure them continuously as long as the combinatorial pseudo triangulation does not change.
Rigidity, Generic Rigidity, In nitesimal Rigidity. A framework is (locally) rigid if its vertices cannot
be moved continuously while preserving the lengths of the edges, except for translations and rotations. Otherwise it is exible. It is in nitesimally exible if there exists an assignment of velocity vectors vi to each vertex pi so that hpi pj ; vi vj i = 0, where h; i is the dot product. Otherwise it is in nitesimally (or rstorder) rigid. A graph is generically rigid if it is rigid for all embeddings on generic sets of points (see the rigidity theory references for precise de nitions). Certain embeddings of generically rigid graphs may be in nitesimally exible, or even exible. See Fig. 7. Notice that these frameworks are cyclic. A graph is minimally rigid if it is rigid and removing some edge creates a graph which is no longer rigid. Laman's theorem provides a combinatorial characterization for minimally rigid graphs on generic embeddings: these are graphs on n vertices with exactly 2n 3 edges, and such that every subset of k vertices spans no more than 2k 3 edges. Henneberg constructions provide an inductive construction of generically rigid graphs via two types of local additions. A step of type 1 involves adding a vertex, joined by two new adjacent edges to two previously constructed vertices. A step of type 2 adds a new vertex and three adjacent egdes to three old vertices such that at least two are joined by an edge, and then drops one of the existing edges among these three old vertices. The reader unfamiliar with these constructions may recognize, in the simplest applications of the type 1 steps which preserve planarity, basic ways of producing triangulations.
(b)
Figure 7: (a) A generically rigid graph in an in nitesimally exible embedding. (b) A rigid framework with a non-generically rigid underlying graph. Two frameworks are combinatorially equivalent if there is a one-to-one correspondence between their vertices preserving edges and faces, the outer face and its orientation, and the underlying partial oriented matroid, i.e. the circular ccw order of lines through the edges around each vertex. This just means that they have the same underlying combinatorial framework (and the embeddings have the same orientation, but we'll skip this detail here). In particular, we de ne equivalent pseudo triangulations. The class of all combinatorially equivalent pseudo triangulations
(a)
(b)
Figure 8: (a) A 1DOF mechanism (Peaucellier's linkage). (b) A framework with an underlying generically minimally exible graph, and with a rigid (but not in nitesimally rigid) embedding.
One-degree-of-freedom mechanisms. A mechanism is an embedded framework which is exible. Its
degree of freedom (DOF) is the dimension of its con guration space (after factorization to translations and rotations). A generic minimally exible graph is a
generically rigid graph with one edge removed (in particular, it has exactly 2n 4 edges). In a generic embedding, a minimally exible graph is a one degree of freedom (1DOF) mechanism, but in other embeddings it may even be rigid, see Fig.8. Notice that these examples are planar but not acyclic. Monotone mechanisms. A 1DOF mechanism is monotone if the non-rigid pairwise interdistances either all increase or all decrease during the local motion. See Fig.6 for examples. E.g the Peaucellier linkage in Fig.8 is not monotone, neither is it acyclic. 3 Rigidity of Pseudo-Triangulations
From now on we are interested only in minimum pseudo triangulations and show that they have a wealth of combinatorial and rigidity theoretical properties.
Theorem 3.1 (Characterization of minimum pseudo triangulations) Let G = (V; E ) be a graph embedded on the set P = fp1 ; ; pn g of points. The following properties are equivalent. 1. G is a minimum pseudo-triangulation. 2. The edges E of G form an acyclic and planar set of segments, and E is maximal with this property (of being both planar and acyclic). 3. G is an acyclic pseudo triangulation of the convex hull of P . 4. (pseudo triangle Laman count) The faces of G are pseudo-triangles and the number of edges is 2n 3. 5. (planar acyclic Laman count) The set of edges E is planar, acyclic and has 2n 3 elements. 6. (planar acyclic Henneberg construction) G can be constructed inductively as follows. Start with a triangle. At each iteration, add a new vertex in one of the faces of the already constructed embedded graph (which will be an acyclic pseudo triangulation). Connect in one of the two ways (see Fig. 9): (a) Type 1: (degree 2) Join the vertex with two tangents to the already constructed part. If the new vertex is outside the convex hull, the two tangents are uniquely de ned. If it is inside an internal pseudo triangular face, there are three di erent ways of adding two tangents to the three inner convex chains of the face. (b) Type 2: (degree 3) Add two tangents as before. Then choose an edge on the convex chain between the two tangent points, remove it. This creates a pseudo 4-gon. Repseudo triangulate by adding the unique bitangent di erent from the one just removed.
Moreover, if any of the above conditions is satis ed, then the subgraph induced on any subset set of k vertices has at most 2k 3 edges (the hereditary property). For the proofs, we will need a series of basic de nitions and facts, which we present in a sketchy manner. Given a point outside a convex hull, a tangent from the point to the hull is a line segment containing all the hull vertices on one side and touching it at a vertex. Given two convex hulls, a bitangent is a line segment touching each hull in one point and whose supporting line does not separate the vertices on the same hull. 1. Given a convex hull and an exterior vertex, there exist exactly two tangents from the point to the hull. 2. Given a pseudo triangle and a vertex interior to it, there exist exactly 3 tangents, all interior to the pseudo triangle, from the point to the (convex hull of the) three inner convex chains. 3. Given a pseudo 4-gon, there exist exactly two ways of adding a bitangent between (the convex hulls of) two inner convex chains. Each one induces a partitioning of the pseudo 4-gon into two pseudo triangles. 4. (Flips in pseudo triangulations) Two adjacent pseudo triangles can be ipped: the unique common edge is deleted and replaced with another one (for which there is a unique choice) to obtain again two adjacent pseudo triangles (see Fig.3(c)).
(a)
(b)
Figure 9: Henneberg steps. (a) type 1 and (b) type 2. Top level, when the new vertex is added on the outside face, bottom level, when it is added inside a pseudo triangular face.
Proof: 1 ! 2 This is the most tedious to prove, so we give only a short sketch here. The proof is by contradiction. Assume G contains an acyclic vertex A. Then A is internal (not on the convex hull). Using the above properties, we argue that there is a sequence of deletions of edges, starting with an edge adjacent to A, and re-pseudo-triangulations of the larger faces thus obtained, which contains fewer edges. 2 ! 1 Assume G is maximally planar and acyclic. We prove by contradiction that if G is not a pseudo triangulation, then we can add edges in an acyclic and
planar fashion, thus contradicting maximality. If G is not connected, add bitangents between the convex hulls of di erent components. If it does not contain its convex hull edges, add them. These operations preserve planarity and acyclicity, hence by maximality we can assume G is both connected and contains the convex hull edges. Similarly, we may argue that its faces are semi-simple polygons. If they are not pseudo triangles, we can always add internal bitangents. 2 ! 3 follows immediately now, and the reverse from an adaptation of the proof of the rst implication. 3 ! 2 is straightforward. 3 ! 5 This is one of the interesting parts. We present here a proof based on a continuous motion argument. Move the points continuously from the original position to convex position. Changes happen at discrete steps, when three points on the same face become collinear. It is easy to show how to locally patch the pseudo triangulation at each event without increasing the number of edges. When all the points reach convex position, the pseudo triangulation becomes a triangulation of a convex set, which has exactly 2n 3 edges. 5 ! 4 follows from 1 and 2. The fact that any subset of k vertices induces at most 2k 3 edges folllows easily, since acyclicity and planarity are hereditary properties (hold on subsets). 5 and 4 ! 6 We work out the construction in reverse. Because of the edge count, a simple counting argument showss that there must exist at least one vertex of degree strictly less than 4. If there exists a vertex of degree 2, its two adjacent edges are tangent to the face obtained by removing them, because of acyclicity. For a vertex of degree 3, the two extreme edges adjacent to it must be tangents (because of acyclicity). The face obtained by removing the third edge is a pseudo 4 gon (follows from the other equivalences), and the addition of the second bitangent recreates a pseudo triangle. Removing the vertex, the remaining graph satis es the same properties (because of the hereditary property). Hence the argument continues. 6 ! 3, 5 and 4 are straightforward: at each step the number of vertices increases by 1 and the number of edges by 2.
their properties and relationship to the convexi cation problem. To illustrate that there is not a unique way for constructing them (because just simply adding edges at random while preserving acyclicity would work), we sketch here two possible constructions for pseudo triangulations of polygons which run in O(n2 ) time. We will use the second one in the complexity analysis. More eÆcient constructions (canonical greedy), running in O(n log n) can be obtained via an adaptation of Pocchiola and Vegter's greedy ip algorithm [29]. We wish to thank Michel Pocchiola for pointing this out. Incremental algorithm. The pseudo triangulation is constructed incrementally, adding one edge of the polygon at a time, in ccw order, starting at a vertex on the convex hull. This insures that the last step will reuse a previously inserted edge and won't necessitate any additional deletions. At each step of the insertion we add a vertex, a polygon edge and one additional edge. However, we might have to displace or modify several other edges, depending on whether the acyclicity condition, the planarity condition, or both are violated by the insertion of the new polygon edge. We show how to modify some of the added pseudotriangulation edges using an argument which we call the rubber band argument with snapping, which locally modi es (in linear time) the edges adjacent to the new polygon edge to preserve acyclicity and planarity. The details are deferred to the full paper.
2
To simplify the terminology, in the rest of this paper we will refer to a minimum or acyclic pseudo triangulation as simply a pseudo-triangulation.
Theorem 3.2 (Algorithms) Any acyclic set of edges can be extended to a pseudo-triangulation by arbitrarily adding edges while preserving acyclicity. A canonical extension can be constructed deterministically in O(n log n) time. If the set of points contains repetitions or collinearities, the resulting faces may be semisimple pseudo triangles. Note that in this paper we have not aimed at proposing the best algorithmic solutions for constructing pseudo triangulations and concentrated instead on
Figure 10: A typical step of the incremental algorithm for computing the pseudo-triangulation of a polygon.
Recursive algorithm. Compute the convex hull of
the polygon. The edges of the convex hull which are not edges of the polygon are subdividing the polygon into pockets. Imagine removing the pockets: what is left is a convex polygon. Triangulate it. Then put
back the pockets and bend the diagonals of the triangulation along shortest (geodesic) paths inside the polygon. The process can now be repeated for each pocket. But the convex hulls of pocket subpolygons may intersect (not along polygon edges, but along some of the added hull edges). Again, we will bend the added convex hull edges to get geodesic convex hulls, then recurse inside each pocket. The construction has the added advantage of being able to count the number n 3 of added edges via a simple charging scheme, whereas an edge is charged to either a geodesic hull of some subpolygon or to a shortest path between two vertices of such a subpolygon. We omit the details here. We associate a weight function to the pseudo triangulation constructed by this algorithm: f (n) = number of bends in all the shortest paths and geodesic hulls of this construction. Since there are linearly many shortest paths and geodesics accounted for (we consider maximal paths only, not subpaths), and each can have at most linearly many bends, f (n) = O(n2 ). When the polygon is convex f (n) = O(n). We will use this function for the complexity analysis.
Theorem 3.3 (Rigidity property of pseudo triangulations) Pseudo-triangulations are in nitesimally rigid (and hence rigid), and minimally so (removing an edge the property no longer holds). The proof is an adaptation, using the properties of pseudo triangulations, of known proofs of Laman's theorem for generic rigidity of graphs with the hereditary (2n 3)-property (every subset of k vertices spans at most 2k 3 edges, and there are 2n 3 edges in total). 4 Monotone 1DOF mechanisms from pseudo triangulations
Theorem 4.1 (Main property of acyclic planar pseudo-triangulations) A rigid bar-and-joint framework whose underlying graph is obtained by removing a convex hull edge from a pseudo-triangulation is a 1DOF monotone mechanism. Proof (sketch). The proof is an extension of the one in [12] (subsequently refered to as the \CDR proof"), using the acyclic property of the edges, and the linear independence in the con guration space of the 2n-4 bars. In particular, we have strengthened their result, by showing that any framework which is generically independent (obtained from a generically rigid (dependent) framework by removing some bars), acyclic and does not contain all the convex hull edges is exible and monotone. We rst show that there exists a unique in nitesimal motion (this is a stronger statement than the one in the CDR proof for polygons, because we restricted the number of degrees of freedom of the mechanism). The proof depends only on the combinatorial type of the pseudo triangulation, hence as long as this does not
change, the motion continues. This step can be made more precise by an argument as in the CDR proof (based on integration of the resulting vector eld), and produces the desired trajectory in con guration space. To prove the existence of the monotone in nitesimal motion, we also have to adapt slightly the argument in the CDR proof, using LP duality, Maxwell's Theorem lifting and the mountain-valley argument of [36]. Add all possible diagonals (\struts") and planarize the graph. By LP duality it suÆces to show that there exists no self-stress which is positive on the struts. For the sake of a contradiction, assume there exists such a self-stress and use Maxwell Theorem to lift the picture in 3d to a polyhedral surface. Then cutting the polyhedral surface with a horizontal plane epsilon below the vertex with maximum z coordinate in the lifting, one obtains a polygon, which will have at least three vertices on the convex hull that must correspond to mountain edges. The projection of the mountain edges on the plane of the section induces three cyclic vectors, which can only be along the edges of the pseudo triangulation. This contradicts the acyclicity of the pseudo-triangulation. See Fig. 11, where M is the vertex where the maximum is attained. Just like in the CDR proof, the argument can be easily extended to the case when the maximum is attained on an edge or face, and acyclicity guarantees that we obtain a contradiction. M f
e
f M
e
a c b
(a)
c
a d
d
b
(b)
Figure 11: The mountain/valley argument: (a) Cutting just below the vertex of maximum z -coordinate, we get the image in (b). Our case needs a few more details added to the basic structure of CDR proof. First, we need the assumption that the rank of the original system of equalities (corresponding to the bars of the framework) is 2n 4, otherwise there will be no basic solutions to the linear program. The argument based on Maxwell lifting is then used to show the existence of a positive solution, however we need extra care, since removing an edge from the framework to get the rank down to the critical value of 2n 4 does not, in itself, imply that there exists a strictly increasing motion on all diagonals despite the fact that the mountain-valley argument holds on the interior points. Indeed, there are both monotone and non-monotone mechanisms arising from pseudo-triangulations by removing an edge, but only those obtained by removing a convex hull edge are monotone. A simple extension of the argument covers the convex hull vertices and semi-simple pseudo triangles. We defer these details to the full paper.
5 The Main Result
Each monotone mechanism induced by a pseudo triangulation can be moved as long as the edge vectors remain acyclic. We complete the proof by showing how to glue together the trajectories corresponding to these motions. A change in acyclicity occurs when two bars become collinear. At that point, we have to recompute a new pseudo triangulation. The following theorem proves that this can be done either with local changes and keeping the same number of vertices in the original polygon, or by decreasing by at least one the number of vertices of the polygon and applying induction.
Theorem 5.1 (Gluing trajectories at alignment events) When two edges align, one of the following cases two can occur: 1. Two adjacent edges of the polygon become collinear. In this case, we freeze the joint, eliminating one vertex of the polygon (and apply induction to continue). 2. Two adjacent added diagonals or one diagonal and an edge of the polygon become collinear. In this case, we perform a ip in the pseudo triangulation to obtain a pseudo triangulation with a semi simple face. An example is depicted in Fig. 12.
Another problem occurs whan we freeze two aligned edges of the polygon. In this case we must get rid of the other diagonals (if any) adjacent to the vertex, which can also be done by local changes (but may involve linearly many edges). Occasionally this operation rigidi es the framework: then we must pick up another convex hull edge to remove from the convex hull. All these details are unproblematic, and the full description is deferred to the full paper.
Theorem 5.2 Termination and complexity analysis The convexi cation of a polygon terminates in nitely many steps, which is at most O(n2 ) steps if we use the weight function based on shortest paths and geodesic hull as invariant. Proof With some care in the patching strategy, it can be shown that no combinatorial pseudo triangulation will occur twice in the convexifying motion. Since there are nitely many combinatorial pseudo triangulations, the algorithm terminates. A
A
B
B C
C
A
B C
Figure 13: When the two aligning edges do not belong to the polygon, there are two possible diagonal ips. A ip reduces the number of bends in geodesic paths using the vertex involved in the alignment.
(a)
(b)
(c)
Figure 12: Patching the pseudo triangulation by a local ip when two bars (not both polygon edges) align. The above theorem is not stated in full generality, to avoid cluttering the overall picture with details. In particular, several vertices may straighten simultaneously, but the same type of argument would work. Alternatively, we can imagine a perturbation argument, used to help with the computation of a new pseudotriangulation at the boundary case. We must avoid using the same two edges that just became collinear. This can be done by perturbing the joint vertex to a nearby position which would intuitively correspond to what the mechanism would look like \right after" passing through the straightened position. It will no longer be acyclic, and there will be exactly one edge whose removal will make it acyclic again. Then we can use the extension theorem for pseudo triangulations to add a new edge. Perturbing back to the original position of the vertex, the induced mechanism will have two overlapping edges and one of its faces will be a semi-simple pseudo triangle (see Fig.12).
A more careful accounting can bring down the number of steps to O(n2 ). We use the weight function introduced for the recursive algorithm in section 3 to keep track of the number of bends in the shortest paths and geodesic hulls. The key observation is that a local ip decreases the number of bends by at least one (or more, depending on how many geodesic paths go through the vertex where the event happened). The special events (when two edges of the polygon align) are only linearly many, and they may not increase the weight function by more than O(n) each (or: recompute a recursive pseudo triangulation on fewer vertices at that point). The nal value of the weight function is at most linear. Hence the number of steps is at most O(n2 ).
Theorem 5.3 Main Result: Convexi cation of Planar Chains with Monotone Motions Every planar polygon can be convexi ed with at most O(n2 ) motions. Each motion is induced by a 1DOF monotone mechanism constructed from a pseudo triangulation with a hull edge removed, which is moved until two of its adjacent edges align. At that point a local
ip of the diagonals restores a pseudo triangulation. The complete trajectory in con guration space is a sequence of simple curves, each one naturally parametrized
by a rotating edge in the work space. A rst pseudo triangulation can be computed eÆciently (O(n log n) or O(n2 )) by several algorithms and updated in (at most) linear time per step. 6 Conclusions
We have shown how to compute algorithmically a trajectory in the con guration space of a planar linkage, corresponding to a motion that convexi es the polygon without producing self-crossings along the way. The proof is based on a number of novel ideas, most prominently the use of pseudo triangulations, whose main combinatorial and rigidity theoretic properties have been described. But still many more await attention, such as: study the graph whose vertices correspond to pseudo triangulations and whose edges correspond to ips in pseudo 4-gons. What is its diameter? How many pseudo triangulations of a point set or polygon are there? Is there any interesting polytope whose 1-skeleton is the graph of pseudo triagulations (as there is for regular triangulations [15])? Since pseudo triangulations can be naturally de ned in the coordinate-free context of oriented matroids, what combinatorial properties would distinguish them (as a collection) from the ones realizable in the euclidian plane? Is there a concept of a regular pseudo triangulation, as it is for triangulations? From the point of view of Rigidity Theory, we have exhibited a class of planar rigid graphs with elegant combinatorial properties. We think that the rigidity theoretic properties of pseudo triangulations will prove to be useful in other applications. It would be interesting to understand the properties of the nonmonotone mechanisms obtained by removing a nonconvex hull edge of a pseudo triangulation and their degenerate con gurations. All the pictures in this paper have been produced using the software package Cinderella [31], which supports motions of 1DOF mechanisms and even animates them. But not all \pseudo-triangulation-minushull-edge" mechanisms admit a Cinderella construction. It would be interesting to classify pseudo triangulations in a complexity hierarchy, based on the extra primitives that should be included in Cinderella to simulate them (i.e. added to basic ruler and compass constructions). This is also related to the problem of computing (in the real RAM model) the coordinates of the points realizing a con guration of a pseudo triangulation mechanism at one edge alignement moment, given the coordinates of the points at the previous event. It probably cannot be done better than using standard numerical approximation techniques. A web page containing Cinderella animations and other graphical and 3d illustrations of our approach can be found at the author's url, http://cs.smith.edu/ streinu. Thanks to my students Beenish Chaudry, Victoria Manfredi, Christine Rice and Elif Tosun for their help.
Acknowledgements This work was possible due to the stimulating resarch environment I encountered at the McGill University Workshops on Folding and Unfolding Problems in Computational Geometry, Barbados 1998 and 2000, the Monte Verita 1999 Discrete and Computational Geometry Workshop in Ascona, Switzerland, and the Geometry Festival, an International Workshop on Discrete Geometry and Rigidity in Budapest, Hungary, Nov. 1999. I am grateful to all the organizers for their invitation (Andras Bezdek, Karoly Bezdek, Karoly Borczy, Bob Connelly, J.E. Goodman, Anna Lubiw, R. Pollack, E. Welzl and Sue Whitesides) and to the many participants (too many to list here, but see [7], [6]) for enthusiastical conversations on this and related problems. I am particularly indebted to Gunter Rote, whose generous sharing of his own ideas (on the use of monotone motions and the linear programming approach to this problem) provided the inspiration for all my subsequent work. I acknowledge useful conversations on various aspects of Rigidity Theory with Bob Connelly, Henri Crapo, Ruth Haas, Brigitte and Herman Servatius and Walter Whiteley. Many thanks to Ciprian Borcea for pointing several connections with the mathematics literature and for stimulating conversations. References
[1] B. Aronov, J.E. Goodman and R. Pollack. Convexifying a Planar polygon in R3 . manuscript, 1999. [2] P. Agarwal, J. Basch, L. Guibas, J. Hershberger and L. Zhang. Deformable Free Space Tilings for Kinetic Collision Detection, Proc. 4th Int. Workshop on Alg. Found. Robotics (WAFR), Dartmouth College, March 2000, [3] J. Basch, J. Erickson, L. Guibas, J. Hershberger and L. Zhang. Kinetic Collision Detection Between Two Simple Polygons, Proc. 10th ACM-SIAM Symp. Discr. Algorithms (SODA), 1999, pp.102111. [4] S. Basu, R. Pollack and M.-F. Roy. Computing Roadmaps of Semi-algebraic Sets on a Variety, JAMS, 13, pp. 55-82, 2000. [5] S. Basu, R. Pollack and M.-F. Roy. Algorithms in Real Algebraic Geometry, book, in preparation. [6] T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O'Rourke, S. Robbins, I. Streinu, G. Toussaint and S. Whitesides. On recon guring tree linkages: Trees can lock, Proc. 10th Canad. Conf. Comp. Geom., Montreal, 1998. [7] T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O'Rourke, M. Overmars, S. Robbins, I. Streinu, G. Toussaint and S. Whitesides. Locked and unlocked polygonal chains in 3D, Proc. 10th
ACM-SIAM Symp. Discrete Algorithms (SODA) 1999, pp. 866-867. [8] A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler. Oriented Matroids, Cambridge University Press, 1993 [9] J. Bokowski. Oriented Matroids, in P.M. Gruber and J.M. Wills, Handbook of Convex Geometry, North Holland, 1993.
[23] M. Kapovich and J. Millson. Universality Theorems for con gurations spaces of planar linkages, preprint, 1998. [24] L. Kavraki, P. Svestka, J.C. Latombe and M. Overmars. Probabilistic roadmaps for path planning in high-dimensional con guration spaces. IEEE Trans.Robot. Automat., 12(4): 566-580, Aug. 1996.
[10] J. Canny. Complexity of Robot Motion Planning, MIT Press, Cambridge, MA, 1988.
[25] R. Kirby. Problems in Low-Dimensional Topology. http://www.math.berkeley.edu/ kirby, 1995.
[11] R. Cocan and J. O'Rourke. Polygonal Chains Cannot Lock in 4d, Proc. 11th Canad.Conf. Comput. Geometry, Vancouver, 1999, pp. 5-8.
[26] W.J. Lenhart and Sue H. Whitesides. Recon guring Closed Polygonal Chains in Euclidian d-space. Discr. Comp. Geometry 13:123-140, 1995.
[12] R. Connelly, E. Demaine and G. Rote. Straightening Polygonal Arcs and Convexifying Polygonal Cycles, this proceedings, 2000
[27] L. Lovasz and Y. Yemini. On generic rigidity in the plane, SIAM J. Algebraic and Discrete Methods, 3, no. 1, 91-98.
[13] H. Crapo and W. Whiteley, Plane self-stresses and projected polyhedra I: The basic pattern, Structural Topology 20, 1993, pp. 55-77.
[28] A. Lubiw and S. Whitesides (org.), McGill University Workshop on Folding and Unfolding, Barbados, 1998, informal proceedings.
[14] P. Erdos. Problem number 3763. Am. Math. Monthly, 46:627, 1935. [15] I. Gelfand, M. Kapranov and A. Zelevinsky. Newton polytopes of principal A-determinants, Soviet Math. Doklady 40, 1990, pp. 278-281.
[29] M. Pocchiola and G. Vegter Topologically Sweeping Visibility Complexes via Pseudo-Triangulations, Discrete Comput. Geometry 16(4), 419-453, Dec. 1996.
[16] J.E. Goodman and R. Pollack. Allowable sequences and Order Types in Discrete and Computational Geometry, in New Trends in Discrete and Computational Geometry, J. Pach (ed.), Springer Verlag, 1993
[30] A. Recski. Matroid Theory and Its Applications, Springer Verlag, 1989. [31] J. Richter-Gebert and U. Kortenkamp. Cinderella, an interactive software package for Geometry, Springer Verlag, 1999.
[17] J. Graver, B. Servatius and H. Servatius. Combinatorial Rigidity, Amer. Math. Soc., Graduate Studies in Mathematics vol.2, 1993.
[32] J. Schwartz and M. Sharir. On the piano mover's problem, II. General techniques for computing topological properties of real algebraic manifolds. Adv. in Appl. Math., 4:298-351, 1983.
[18] L. Han and N. Amato A Kinematics-Based Probabilistic Roadmap Method for Closed Chain Systems. Proc. 4th Int. Workshop on Alg. Found. Robotics (WAFR), Dartmouth College, March 2000.
[33] I. Streinu. Clusters of Stars, Proc. ACM Symp. Comp. Geometry, Nice, 1997, pp. 439-441.
[19] J. Hopcroft, D. Joseph and S.H. Whitesides. On the movement of robot arms in 2-dimensional bounded regions, SIAM J. Comput., 14:315-333, 1985.
[34] G. Toussaint. The Erdos-Nagy Theorem and its Rami cations, Proc. Canad. Conf. Comp. Geom., 1999. [35] W. Whiteley. Motions and Stresses of Projected Polyhedra, Structural Topology 7, pp. 13-38, 1982. Amer. Math. Soc. 1996.
[20] D. Jordan and M. Steiner. Con guration Spaces of Mechanical Linkages, Discrete Comput. Geometry 22, pp. 297-315, 1999.
[36] W. Whiteley. Some Matroids from Discrete Applied Geometry, Contemporary Mathematics, vol. 197, pp. 171-311, Amer. Math. Soc. 1996.
[21] D. Joseph and W.H. Platinga. On the complexity of reachability and motion planning questions, Proc. Symp. Comp. Geom., pp. 62-66, 1985.
[37] W. Whiteley. Rigidity and Scene Analysis, in Handbook of Discrete and Computational Geometry, (J.E. Goodman, J. O'Rourke, eds.), 1997, pp. 893-916.
[22] M. Kapovich and J. Millson. On the moduli spaces of polygons in the Euclidian plane, Journ. of Di . Geometry, vol. 42, N 1, 1995, pp. 133-164.
[38] S.H. Whitesides. Algorithmic issues in the geometry of planar linkage movement, Austral. Comp. J., Spec. Issue on Algorithms, 24(2):42-50, 1992.
